We extend Jarzynski's work relation and the second law of thermodynamics to a heat conducting system, which is operated by an external agent. These extensions contain a nonequilibrium contribution expressed as the violation of the (linear) response relation caused by the operation. We find that a natural extension of the minimum work principle involves information about the time-reversed operation, and is far from straightforward. Our work relation may be tested experimentally especially when the temperature gradient is small.
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